I mean right?!
I love the idea of different animated spellcasting gestures or somatic flair.
Another cool idea might be to use a different dead language for the verbal incantations.
Right now Latin is used for basically everything, but what if it was like Sanskrit for Wizards and Latin for Priests, Celtic for Druids? Or you know something, like that. Most users probably wouldn't register these subtle vocalization differences, since they're usually short phrases, but that might also have a cumulative differentiating effect.
Just little things like what Starblaireau suggested about using the regalia or sigils to distinguish one Cleric from Another. Or Argyle's idea with the halos.
Another way in would be through the actual story. I still remember on my first playthrough, I selected Warlock for the MC just to check it out, and I thought Raphael was class specific. That was just the coolest shit ever, until I played again as a Fighter and had the exact same Raphael scene play out. I think they could give class flavor if they approached lvl1 more like the old "Complete guide to [insert class X]" where in order to advance out of the prologue the player first has some kind of unique class specific scene-setter play out. I think the time to do that is when the worm is put in our eyes, before the Nautiloid. That'd be a great point to have like a priestly "Vision" occur that sort of sets the stage. Like memory flashbacks or something, that changes depending on ones chosen class.